Math Problem Statement
Solution
The problem presented involves a normal distribution of rice bag weights. Here's a breakdown:
- The weights are normally distributed with mean and standard deviation .
- 97.5% of the bags weigh more than 30.5 kg.
- 88.49% of the bags weigh less than 46.3 kg.
- We are tasked with finding the values of and .
Step 1: Use of Z-scores
From the standard normal distribution table (or a calculator), we can convert the given percentages into Z-scores:
-
97.5% of the bags weigh more than 30.5 kg:
- This means 2.5% weigh less than 30.5 kg.
- From the Z-table, the Z-score corresponding to 2.5% in the lower tail is approximately -1.96.
-
88.49% of the bags weigh less than 46.3 kg:
- From the Z-table, the Z-score corresponding to 88.49% is approximately 1.19.
Step 2: Set up the equations
Using the Z-score formula:
where is the data value, is the mean, and is the standard deviation.
-
For the 30.5 kg bag:
This gives the equation: or -
For the 46.3 kg bag:
This gives the equation:
Step 3: Solve the system of equations
We now solve the two equations:
Substitute Equation 1 into Equation 2: Simplify:
Step 4: Find
Substitute into Equation 1:
Final Answer:
Would you like further details or clarifications?
Follow-up Questions:
- What is the significance of the Z-score in the context of this problem?
- How do we interpret the mean and standard deviation in real-life scenarios?
- Can you explain why the Z-score table was used and its importance?
- What would happen if we changed the percentage from 97.5% to 99%?
- How would you solve this problem if only one percentage was given?
Tip:
Always ensure you know how to read and use the Z-table for solving normal distribution problems like this one.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Normal Distribution
Z-scores
Statistics
Formulas
Z = (X - μ) / σ
Theorems
Normal Distribution Theorem
Z-score Interpretation
Suitable Grade Level
Grades 10-12
Related Recommendation
Calculate Mean and Standard Deviation of Normal Distribution with Given Percentages
Statistical Analysis of Weight Data from 80 Students
Statistics Problem: Sample Mean and Probability in Normal Distribution
Find Mean, Standard Deviation, and Percent Over 2 kg in Normally Distributed Potatoes
Normal Distribution Probability for Sample and Population - Mean μ = 33.2, Standard Deviation σ = 68.9